This doesn't just include hospitals and doctor's offices, but medical equipment suppliers and some pharmacies, too.
They will ask you to fill out a form for your household income, but you do not have to include people who are not legally responsible for your bill. You should include dependents, though, because they contribute to the household size, which effects how much assistance you get. They will ask you for proof of income which can include pay stubs, a copy of social security, disability, or other benefits, a W-2, a copy of your bank account, a federal income tax return, basically anything that shows any type of income. If you don't have a bank account, make sure they know. They may need some other form to replace it.
They may also need proof of residency like a utility bill, mortgage coupon, or rental agreement and proof of identity like a driver's license, social security card, birth certificate, or passport. If you do not have a home, they may assign you a social worker to work something out (in the case of a larger facility like a hospital.)
This is based on my experience as a person working in the healthcare field and may not reflect all requirements or experiences. Please contribute your experience and knowledge to this thread.
Edit: I forgot to mention, you will probably have to update your financial status every year, so make sure to call your provider about it after that period to see if you need to.